270 THE THEORY OF SCREWS. [254- 



The transformation having been effected, an important result is im 

 mediately deduced. Let the transformed function be denoted by 



then we have 



ft=- 

 PI 



PI 



whence it appears that the six screws of reference are the common screws of 

 the two systems. We thus find that in this special case of homography 



The six common screws of the two systems are co-reciprocal. 



The correspondence between impulsive screws and instantaneous screws 

 is a particular case of the type here referred to. The six common screws of 

 the two systems are therefore what we have called the principal screws of 

 inertia, and they are co-reciprocal. 



255. Correspondence of a Screw and a system. 



We shall sometimes have cases in which each screw of a system cor 

 responds not to a single screw but to a system of screws. For the sake of 

 illustration, suppose the case of a quiescent rigid body with two degrees of 

 freedom and let this receive an impulsive wrench on some screw situated 

 anywhere in space. The movement which the body can accept is limited. 

 It can, indeed, only twist about one of the singly infinite number of screws, 

 which constitute a cylindroid. To any screw in space will correspond one 

 screw on the cylindroid. But will it be correct to say, that to one screw on 

 the cylindroid corresponds one screw in space ? The fact is, that there are 

 a quadruply infinite number of screws, an impulsive wrench on any one of 

 which will make the body choose the same screw on the cylindroid for its 

 instantaneous movement. The relation of this quadruply infinite group is 

 clearly exhibited in the present theory. It is shown in 128 that, given a 

 screw a on the cylindroid, there is, in general one, but only one screw 6 on 

 the cylindroid, an impulsive wrench on which will make the body commence 

 to twist about a. It is further shown that any screw whatever which fulfils 

 the single condition of being reciprocal to a single specified screw on the 

 cylindroid possesses the same property. The screws corresponding to a thus 

 form a five-system. The correspondence at present before us may therefore 

 be enunciated in the following general manner. 



To one screw in space corresponds one screw on the cylindroid, and to one 

 screw on the cylindroid corresponds a five-system in space. 



